Traveling waves for a microscopic model of traffic flow

Wen Shen, Karim Shikh-Khalil

Research output: Contribution to journalArticlepeer-review

10 Scopus citations


We consider the follow-the-leader model for traffic flow. The position of each car zi(t) satisfies an ordinary differential equation, whose speed depends only on the relative position zi+1(t) of the car ahead. Each car perceives a local density ?i(t). We study a discrete traveling wave profile W(x) along which the trajectory (?i(t), zi(t)) traces such that W(zi(t)) = ?i(t) for all i and t > 0; see definition 2.2. We derive a delay differential equation satisfied by such profiles. Existence and uniqueness of solutions are proved, for the two-point boundary value problem where the car densities at x ? ±? are given. Furthermore, we show that such profiles are locally stable, attracting nearby monotone solutions of the follow-the-leader model.

Original languageEnglish (US)
Pages (from-to)2571-2589
Number of pages19
JournalDiscrete and Continuous Dynamical Systems- Series A
Issue number5
StatePublished - May 2018

All Science Journal Classification (ASJC) codes

  • Analysis
  • Discrete Mathematics and Combinatorics
  • Applied Mathematics


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